Today’s Puzzle:

If you can print a copy of the puzzle from this Excel sheet, 10 Factors1773-1785, it will look like this:

Note: I have revised this puzzle since originally publishing it. I was horrified to discover that the original puzzle had two solutions. I apologize for any inconvenience I may have caused. This revised puzzle only has one solution.

Factors of 1785:

17 × 5 = 85, so 1785 is divisible by 17.

• 1785 is a composite number.
• Prime factorization: 1785 = 3 × 5 × 7 × 17.
• 1785 has no exponents greater than 1 in its prime factorization, so √1785 cannot be simplified.
• The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1785 has exactly 16 factors.
• The factors of 1785 are outlined with their factor pair partners in the graphic below.

More About the Number 1785:

Did you notice that 3, 5, 7, and 357 are all factors of 1785?
Or that 35 and 51 make a factor pair, and 3, 5, and 1 are also factors?

1785 is the hypotenuse of FOUR Pythagorean triples:
273-1764-1785
756-1617-1785
840-1575-1785
1071-1428-1785

1785 is the difference of two squares in EIGHT different ways:
893² – 892² = 1785,
299² – 296² = 1785,
181² – 176² = 1785, and five more ways. Can you find them?

1785 is a Palindrome in a couple of bases:
It’s 123321 in base 4, because 1(1024) + 2(256) + 3(64) + 3(16) + 2(4) + 1(1) = 1785.
And it’s 3F3 in base 22, because 3(22²) + 15(22) + 3(1) = 1785.